Standard Step Method

The Standard Step Method (STM) is a computationaltechnique utilized to estimate one-dimensional surfacewater proles in open channels with gradually varied owunder steady state conditions. It uses a combination of theenergy, momentum, and continuity equations to deter-mine water depth with a given a friction slope (Sf ) , chan-nel slope (S0) , channel geometry, and also a given owrate. In practice, this technique is widely used throughthe computer program HEC-RAS, developed by the USArmy Corps of Engineers Hydrologic Engineering Cen-ter (HEC).[1]

1 Open Channel Flow Fundamen-tals

Figure 1. Conceptual gure used to dene terms in the energyequation.[2]

Figure 2. A diagram showing the relationship for ow depth (y)and total Energy (E) for a given ow (Q). Note the location ofcricital ow, subcritical ow, and supercritical ow.

The energy equation used for open channel ow com-putations is a simplication of the Bernoulli Equation(See Bernoulli Principle), which takes into account pres-sure head, elevation head, and velocity head. (Note, en-ergy and head are synonymous in Fluid Dynamics. See

Pressure head for more details.) In open channels, it isassumed that changes in atmospheric pressure are negligi-ble, therefore the pressure head term used in BernoullisEquation is eliminated. The resulting energy equation isshown below:

H = z + y + v2

2g Equation 1

For a given ow rate and channel geometry, there is arelationship between ow depth and total energy. Thisis illustrated below in the plot of energy vs. ow depth,widely known as an E-y diagram. In this plot, the depthwhere theminimum energy occurs is known as the criticaldepth. Consequently, this depth corresponds to a FroudeNumber (Fn) of 1. Depths greater than critical depthare considered subcritical and have a Froude Numberless than 1, while depths less than critical depth are con-sidered supercritical and have Froude Numbers greaterthan 1. (For more information, see Dimensionless Spe-cic Energy Diagrams for Open Channel Flow.)

Fn =v

(g AB )0:5 Equation 2

Under steady state ow conditions (e.g. no ood wave),open channel ow can be subdivided into three types ofow: uniform ow, gradually varying ow, and rapidlyvarying ow. Uniform ow describes a situation whereow depth does not change with distance along the chan-nel. This can only occur in a smooth channel that doesnot experience any changes in ow, channel geometry,roughness or channel slope. During uniform ow, theow depth is known as normal depth (yn). This depthis analogous to the terminal velocity of an object in freefall, where gravity and frictional forces are in balance(Moglen, 2013).[3] Typically, this depth is calculated us-ing the Manning formula. Gradually varied ow occurswhen the change in ow depth per change in ow dis-tance is very small. In this case, hydrostatic relationshipsdeveloped for uniform ow still apply. Examples of thisinclude the backwater behind an in-stream structure (e.g.dam, sluice gate, weir, etc.), when there is a constric-tion in the channel, and when there is a minor changein channel slope. Rapidly varied ow occurs when thechange in ow depth per change in ow distance is sig-nicant. In this case, hydrostatics relationships are notappropriate for analytical solutions, and continuity of mo-mentum must be employed. Examples of this includelarge changes in slope like a spillway, abrupt constric-tion/expansion of ow, or a hydraulic jump.

1

2 3 STANDARD STEP METHOD CALCULATION

2 Water Surface Proles (Gradu-ally Varied Flow)

Typically, the STM is used to develop surface water pro-les, or longitudinal representations of channel depth,for channels experiencing gradually varied ow. Thesetransitions can be classied based on reach condition(mild or steep), and also the type of transition beingmade. Mild reaches occur where normal depth is sub-critical (yn > yc) while steep reaches occur where normaldepth is supercritical (yn

33.1 Newton Raphson Numerical Method

Computer programs like excel contain iteration or goalseek functions that can automatically calculate the actualdepth instead of manual iteration.

3.2 Conceptual Surface Water Proles(Sluice Gate)

Figure 4 illustrates the dierent surface water proles as-sociated with a sluice gate on a mild reach (top) and asteep reach (bottom). Note, the sluice gate induces achoke in the system, causing a backwater prole justupstream of the gate. In the mild reach, the hydraulicjump occurs downstream of the gate, but in the steepreach, the hydraulic jump occurs upstream of the gate. Itis important to note that the gradually varied ow equa-tions and associated numerical methods (including the

Figure 4. Illustration of surface water proles associated with asluice gate in a mild reach (top) and a steep reach (bottom).

standard step method) cannot accurately model the dy-namics of a hydraulic jump.[6] See the Hydraulic jumpsin rectangular channels page for more information. Be-low, an example problem will use conceptual models tobuild a surface water prole using the STM.

4 Example Problem

Solution

4 4 EXAMPLE PROBLEM

Using Figure 3 and knowledge of the upstream and down-

5stream conditions and the depth values on either side ofthe gate, a general estimate of the proles upstream anddownstream of the gate can be generated. Upstream, thewater surface must rise from a normal depth of 0.97 m to9.21m at the gate. The only way to do this on amild reachis to follow an M1 prole. The same logic applies down-stream to determine that the water surface follows an M3prole from the gate until the depth reaches the conju-gate depth of the normal depth at which point a hydraulicjump forms to raise the water surface to the normal depth.Step 4: Use the Newton Raphson Method to solve theM1 and M3 surface water proles. The upstream anddownstream portions must be modeled separately withan initial depth of 9.21 m for the upstream portion, and0.15 m for the downstream portion. The downstreamdepth should only be modeled until it reaches the conju-gate depth of the normal depth, at which point a hydraulicjump will form. The solution presented explains how tosolve the problem in a spreadsheet, showing the calcu-lations column by column. Within Excel, the goal seekfunction can be used to set column 15 to 0 by changingthe depth estimate in column 2 instead of iterating man-ually.

Table 1: Spreadsheet of Newton Raphson Method ofdownstream water surface elevation calculationsStep 5: Combine the results from the dierent prolesand display.

6 4 EXAMPLE PROBLEM

Normal depth was achieved at approximately 2,200 me-ters upstream of the gate.Step 6: Solve the problem in the HEC-RAS ModelingEnvironment:It is beyond the scope of this Wikipedia Page to explainthe intricacies of operating HEC-RAS. For those inter-ested in learning more, the HEC-RAS users manual isan excellent learning tool and the program is free to thepublic.The rst two gures below are the upstream and down-stream water surface proles modeled by HEC-RAS.There is also a table provided comparing the dierencesbetween the proles estimated by the two dierent meth-ods at dierent stations to show consistency between thetwo methods. While the two dierent methods modeledsimilar water surface shapes, the standard step methodpredicted that the ow would take a greater distance toreach normal depth upstream and downstream of the gate.This stretching is caused by the errors associated with as-suming average gradients between two stations of interestduring our calculations. Smaller dx values would reducethis error and produce more accurate surface proles.

The HEC-RAS model calculated that the water backsup to a height of 9.21 meters at the upstream side ofthe sluice gate, which is the same as the manually calcu-lated value. Normal depth was achieved at approximately1,700 meters upstream of the gate.HEC-RAS modeled the hydraulic jump to occur 18 me-ters downstream of the sluice gate.

75 References[1] USACE. HEC-RAS Version 4.1 Users Manual. Hy-

drologic Engineering Center, Davis, CA.

[2] Chaudhry, M.H. (2008). Open-Channel Flow. NewYork:Springer.

[3] Moglen, G. Lecture Notes from CEE 4324/5894: OpenChannel Flow, Virginia Tech. Retrieved April 24, 2013.

[4] Chow, V.T. (1959). Open-Channel Hydraulics. NewYork: McGraw-Hill.

[5] Chaudhry, M.H. (2008). Open-Channel Flow. NewYork:Springer.

[6] Chaudhry, M.H. (2008). Open-Channel Flow. NewYork:Springer.

8 6 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

6 Text and image sources, contributors, and licenses6.1 Text

Standard Step Method Source: https://en.wikipedia.org/wiki/Standard_Step_Method?oldid=596281855 Contributors: Wavelength,Fram, LionMans Account, Yobot, BattyBot, NateJonesAR, Jricheson and Anonymous: 2

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Open Channel Flow Fundamentals Water Surface Profiles (Gradually Varied Flow) Standard Step Method Calculation Newton Raphson Numerical Method Conceptual Surface Water Profiles (Sluice Gate)

Example Problem References Text and image sources, contributors, and licensesTextImagesContent license